Counterexamples to the Corsten-Frankl conjecture on diameter-Ramsey simplices

Abstract: Corsten and Frankl conjectured that a simplex is diameter-Ramsey if and only if its circumcenter lies in its convex hull. We disprove this conjecture in every dimension $d\ge 3$. The main tool is a sufficient criterion based on a higher-order deficit decomposition: if the squared deficits $D^2-|p_i-p_j|^2$ admit a nonnegative decomposition over subsets of the vertex set, with total mass at most $D^2$, then the simplex is diameter-Ramsey. The pairwise deficit criterion of Frankl--Pach--Reiher--Rödl is recovered as a special case. As an application, for every $d\ge 3$ we construct a diameter-Ramsey $d$-simplex whose circumcenter lies outside its convex hull. A particularly simple family has squared edge lengths $|p_1-p_2|^2=|p_1-p_j|^2=7~ (4\le j\le d+1)$, $|p_1-p_3|^2=4$, and $|p_i-p_j|^2=4 ~ (2\le i<j\le d+1)$.

• The paper demonstrates that the conventional circumcenter criterion fails to characterize diameter-Ramsey simplices in dimensions d ≥ 3.
• It introduces a novel higher-order deficit decomposition method to certify the diameter-Ramsey property despite the circumcenter lying outside the simplex.
• Constructive counterexamples and barycentric analysis highlight that nuanced distance distributions offer a more reliable criterion than simple geometric embeddings.

Disproof of the Corsten-Frankl Conjecture on Diameter-Ramsey Simplices

Introduction

This paper addresses a foundational question in Euclidean Ramsey theory regarding the combinatorial and geometric characterization of diameter-Ramsey simplices. Specifically, it disproves the conjecture—formulated by Corsten and Frankl—that a simplex is diameter-Ramsey if and only if its circumcenter lies within its convex hull. The research demonstrates the failure of this conjecture in all dimensions $d \geq 3$ by constructing explicit counterexamples and developing a more general sufficient criterion for the diameter-Ramsey property based on higher-order deficit decompositions.

Background and Definitions

Diameter-Ramsey sets, introduced by Frankl, Pach, Reiher, and Rödl, are finite point configurations in Euclidean space with the property that, for every $q \in \mathbb{N}$, there exists a finite ambient set $R$ of the same diameter as the given set, such that every $q$-coloring of $R$ contains a monochromatic isometric copy of the original configuration. The concept plays a central role in high-dimensional combinatorial geometry and geometric Ramsey theory.

The previous conjecture of Corsten and Frankl posited that a simplex’s diameter-Ramsey property is precisely coded by the location of its circumcenter with respect to its convex hull. This conjecture, if true, would provide a geometric descriptor for a subtle Ramsey-theoretic property.

Main Theoretical Results

Higher-Order Deficit Criterion

The central technical contribution is a sufficient criterion for the diameter-Ramsey property, generalizing the pairwise deficit condition of Frankl-Pach-Reiher-Rödl. Given a simplex $A = \{p_1, \dots, p_n\}$ of diameter $D$, one considers the pairwise deficits $d_{ij} = D^2 - \|p_i - p_j\|^2$. The new criterion posits that if these deficits admit a nonnegative decomposition over subsets $B$ of the vertex set (with total mass at most $D^2$), then $q \in \mathbb{N}$0 is diameter-Ramsey. This generalization is not merely technical, but provides greater flexibility in constructing diameter-Ramsey sets and enables the construction of counterexamples.

Construction of Counterexamples

For every $q \in \mathbb{N}$1, the paper constructs a parameterized family of $q \in \mathbb{N}$2-simplices $q \in \mathbb{N}$3 determined by explicit formulas for their squared edge lengths. Careful selection of the parameters (e.g., $q \in \mathbb{N}$4) yields simplices whose circumcenter lies outside their convex hull, yet the higher-order deficit criterion certifies their diameter-Ramsey property. The construction involves assembling the simplex inside a Cartesian product of regular simplices, ensuring congruence and compatibility of distances.

Barycentric Analysis

By computing barycentric coordinates of the circumcenter within the constructed simplices, it is shown algebraically that one barycentric coordinate can become negative depending on parameter choices. When this occurs, the circumcenter is exterior to the simplex's convex hull. The derivation confirms the existence and explicit form of diameter-Ramsey simplices that violate the conjectured geometry-coloring correspondence.

Contradiction to Established Criteria

The constructed family also evades the established pairwise deficit criterion. For the chosen parameter values, the aggregate pairwise deficit exceeds the squared diameter, meaning the older criterion is not applicable; yet the simplex satisfies the higher-order decomposition and is thus diameter-Ramsey. This establishes strict separation between the power of the new approach and prior sufficient conditions.

Implications and Future Directions

Theoretical Impact

The main theoretical implication is the negation of the circumcenter-based geometric conjecture as a characterization of diameter-Ramsey simplices in high dimensions. The location of the circumcenter is shown to be an insufficient invariant, compelling a search for strictly combinatorial or metric properties, such as those encoded by higher-order deficit decompositions, to classify diameter-Ramsey simplices.

Practical Consequences

While the research is highly structural and abstract, it clarifies that criteria for diameter-Ramsey properties cannot rely on straightforward geometric embedding features, and instead require deeper analysis of distance distributions encoded in the simplex. This has downstream implications for geometric Ramsey theory, especially for constructions and algorithms reliant on diameter-Ramsey certificates.

Open Problems and Prospects

Notably, the methods and results do not address the planar case ($q \in \mathbb{N}$5), for which the correspondence between circumcenter position and the diameter-Ramsey property may still hold. The sufficiency and necessity of higher-order deficit decompositions for arbitrary finite configurations remain open, suggesting possible lines of investigation. The techniques introduced may spur further work in developing algorithmic checks for the diameter-Ramsey property beyond pairwise metrics.

Conclusion

This paper establishes the failure of the Corsten-Frankl conjecture in all dimensions $q \in \mathbb{N}$6 by providing explicit counterexamples: diameter-Ramsey simplices whose circumcenter lies outside the convex hull. The main tool, a higher-order deficit decomposition criterion, strictly generalizes previous sufficient conditions and characterizes a broader class of diameter-Ramsey simplices, independent of circumcenter location. The results indicate that the diameter-Ramsey property is more subtle than previously believed and suggest the importance of analytic deficit decompositions in understanding Ramsey-theoretic phenomena in Euclidean spaces.

Reference: "Counterexamples to the Corsten-Frankl conjecture on diameter-Ramsey simplices" (2604.19126).